In-class Ex 7

Author

Stephen Tay

Published

October 14, 2024

Modified

October 16, 2024

1. Overview

In this in-class exercise, we build a hedonic pricing model for condominiums using Geographically Weighted Regression (GWR) to account for spatial non-stationarity, where relationships between variables differ across geographic locations. This exercise builds on hands-on exercise 7, with additional emphasis on other R packages relevant to GWR

pacman::p_load(olsrr, ggstatsplot, ggpubr, sf, spdep, GWmodel, tmap, tidyverse, gtsummary, performance, see, sfdep)

2. Importing & Transforming Data

We will import and work with two datasets:

  • URA’s 2014 master plan subzone boundary
  • Condominium resale prices 2015

We use st_transform() to assign the correct ESPG code 3414.

mpsz <- st_read(dsn = "data/geospatial", layer = "MP14_SUBZONE_WEB_PL") %>%
  st_transform(3414)
Reading layer `MP14_SUBZONE_WEB_PL' from data source 
  `/Users/stephentay/stephentay/ISSS626-Geospatial-Analytics/In-class_Ex/In-class_Ex07/data/geospatial' 
  using driver `ESRI Shapefile'
Simple feature collection with 323 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension:     XY
Bounding box:  xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
Projected CRS: SVY21

We import the condo resale prices dataset using read_csv().

condo_resale = read_csv("data/aspatial/Condo_resale_2015.csv")
head(condo_resale)
# A tibble: 6 × 23
  LATITUDE LONGITUDE POSTCODE SELLING_PRICE AREA_SQM   AGE PROX_CBD
     <dbl>     <dbl>    <dbl>         <dbl>    <dbl> <dbl>    <dbl>
1     1.29      104.   118635       3000000      309    30     7.94
2     1.33      104.   288420       3880000      290    32     6.61
3     1.31      104.   267833       3325000      248    33     6.90
4     1.31      104.   258380       4250000      127     7     4.04
5     1.32      104.   467169       1400000      145    28    11.8 
6     1.31      104.   466472       1320000      139    22    10.3 
# ℹ 16 more variables: PROX_CHILDCARE <dbl>, PROX_ELDERLYCARE <dbl>,
#   PROX_URA_GROWTH_AREA <dbl>, PROX_HAWKER_MARKET <dbl>,
#   PROX_KINDERGARTEN <dbl>, PROX_MRT <dbl>, PROX_PARK <dbl>,
#   PROX_PRIMARY_SCH <dbl>, PROX_TOP_PRIMARY_SCH <dbl>,
#   PROX_SHOPPING_MALL <dbl>, PROX_SUPERMARKET <dbl>, PROX_BUS_STOP <dbl>,
#   NO_Of_UNITS <dbl>, FAMILY_FRIENDLY <dbl>, FREEHOLD <dbl>,
#   LEASEHOLD_99YR <dbl>

Since the CSV file contains latitude and longitude coordinates, but the coordinate system is unknown, we make an informed guess and assign the closest geographic coordinate system—EPSG 4326. (You could verify the accuracy by plotting the data to ensure the spatial points align with real-world locations). If confirmed, we transform the data to the projected coordinate system EPSG 3414.

condo_resale_sf = condo_resale %>%
  st_as_sf(coords = c("LONGITUDE", "LATITUDE"), 
           crs=4326) %>% # use the geo coordinate system of the lat/long in the csv file
  st_transform(crs=3414)
head(condo_resale_sf)
Simple feature collection with 6 features and 21 fields
Geometry type: POINT
Dimension:     XY
Bounding box:  xmin: 22085.12 ymin: 29951.54 xmax: 41042.56 ymax: 34546.2
Projected CRS: SVY21 / Singapore TM
# A tibble: 6 × 22
  POSTCODE SELLING_PRICE AREA_SQM   AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE
     <dbl>         <dbl>    <dbl> <dbl>    <dbl>          <dbl>            <dbl>
1   118635       3000000      309    30     7.94          0.166            2.52 
2   288420       3880000      290    32     6.61          0.280            1.93 
3   267833       3325000      248    33     6.90          0.429            0.502
4   258380       4250000      127     7     4.04          0.395            1.99 
5   467169       1400000      145    28    11.8           0.119            1.12 
6   466472       1320000      139    22    10.3           0.125            0.789
# ℹ 15 more variables: PROX_URA_GROWTH_AREA <dbl>, PROX_HAWKER_MARKET <dbl>,
#   PROX_KINDERGARTEN <dbl>, PROX_MRT <dbl>, PROX_PARK <dbl>,
#   PROX_PRIMARY_SCH <dbl>, PROX_TOP_PRIMARY_SCH <dbl>,
#   PROX_SHOPPING_MALL <dbl>, PROX_SUPERMARKET <dbl>, PROX_BUS_STOP <dbl>,
#   NO_Of_UNITS <dbl>, FAMILY_FRIENDLY <dbl>, FREEHOLD <dbl>,
#   LEASEHOLD_99YR <dbl>, geometry <POINT [m]>

3. Building the Hedonic Price Model

The following steps are taken to build the hedonic price model.

3.1 Correlation Analysis

To prevent multicollinearity, it is essential to examine relationships between variables to identify those with high correlations. While immediate removal isn’t necessary, these variables should be closely monitored during the multicollinearity test later.

As an alternative to the corrplot package, the ggcorrmat() function from the ggstatsplot package can also be used for correlation analysis.

ggcorrmat(condo_resale[, 5:23])

3.2 Initial Hedonic Pricing Model Using MLR

We begin by building an initial MLR model with all variables deemed relevant for predicting the selling price. We will assess the model in the next step.

condo_mlr <- lm(formula = SELLING_PRICE ~ AREA_SQM + AGE    + 
                  PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE +
                  PROX_URA_GROWTH_AREA + PROX_HAWKER_MARKET + PROX_KINDERGARTEN + 
                  PROX_MRT  + PROX_PARK + PROX_PRIMARY_SCH + 
                  PROX_TOP_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_SUPERMARKET + 
                  PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD + 
                  LEASEHOLD_99YR, 
                data=condo_resale_sf)
summary(condo_mlr)

Call:
lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + PROX_CHILDCARE + 
    PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_HAWKER_MARKET + 
    PROX_KINDERGARTEN + PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + 
    PROX_TOP_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_SUPERMARKET + 
    PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD + 
    LEASEHOLD_99YR, data = condo_resale_sf)

Residuals:
     Min       1Q   Median       3Q      Max 
-3471036  -286903   -22426   239412 12254549 

Coefficients:
                      Estimate Std. Error t value Pr(>|t|)    
(Intercept)           543071.4   136210.9   3.987 7.03e-05 ***
AREA_SQM               12688.7      370.1  34.283  < 2e-16 ***
AGE                   -24566.0     2766.0  -8.881  < 2e-16 ***
PROX_CBD              -78122.0     6791.4 -11.503  < 2e-16 ***
PROX_CHILDCARE       -333219.0   111020.3  -3.001 0.002734 ** 
PROX_ELDERLYCARE      170950.0    42110.8   4.060 5.19e-05 ***
PROX_URA_GROWTH_AREA   38507.6    12523.7   3.075 0.002147 ** 
PROX_HAWKER_MARKET     23801.2    29299.9   0.812 0.416739    
PROX_KINDERGARTEN     144098.0    82738.7   1.742 0.081795 .  
PROX_MRT             -322775.9    58528.1  -5.515 4.14e-08 ***
PROX_PARK             564487.9    66563.0   8.481  < 2e-16 ***
PROX_PRIMARY_SCH      186170.5    65515.2   2.842 0.004553 ** 
PROX_TOP_PRIMARY_SCH    -477.1    20598.0  -0.023 0.981525    
PROX_SHOPPING_MALL   -207721.5    42855.5  -4.847 1.39e-06 ***
PROX_SUPERMARKET      -48074.7    77145.3  -0.623 0.533273    
PROX_BUS_STOP         675755.0   138552.0   4.877 1.20e-06 ***
NO_Of_UNITS             -216.2       90.3  -2.394 0.016797 *  
FAMILY_FRIENDLY       142128.3    47055.1   3.020 0.002569 ** 
FREEHOLD              300646.5    77296.5   3.890 0.000105 ***
LEASEHOLD_99YR        -77137.4    77570.9  -0.994 0.320192    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 755800 on 1416 degrees of freedom
Multiple R-squared:  0.652, Adjusted R-squared:  0.6474 
F-statistic: 139.6 on 19 and 1416 DF,  p-value: < 2.2e-16

3.3 Model Assessment (using olsrr)

In model assessment, we evaluate the model’s p-value and R² to determine overall significance and explanatory power. Next, we assess individual variables to identify any that are non-significant, as these should be removed to enhance the model’s robustness.

The olsrr package is used to generate a comprehensive, tidy report for assessing the MLR model. The summary report indicates that the model is statistically significant (p < 0.05) and explains 64.7% of the variance in the dependent variable. However, since some independent variables are not statistically significant, they should be excluded from the final model.

ols_regress(condo_mlr)
                                Model Summary                                 
-----------------------------------------------------------------------------
R                            0.807       RMSE                     750537.537 
R-Squared                    0.652       MSE                571262902261.220 
Adj. R-Squared               0.647       Coef. Var                    43.160 
Pred R-Squared               0.637       AIC                       42971.173 
MAE                     412117.987       SBC                       43081.835 
-----------------------------------------------------------------------------
 RMSE: Root Mean Square Error 
 MSE: Mean Square Error 
 MAE: Mean Absolute Error 
 AIC: Akaike Information Criteria 
 SBC: Schwarz Bayesian Criteria 

                                     ANOVA                                       
--------------------------------------------------------------------------------
                    Sum of                                                      
                   Squares          DF         Mean Square       F         Sig. 
--------------------------------------------------------------------------------
Regression    1.515738e+15          19        7.977571e+13    139.648    0.0000 
Residual      8.089083e+14        1416    571262902261.220                      
Total         2.324647e+15        1435                                          
--------------------------------------------------------------------------------

                                               Parameter Estimates                                                
-----------------------------------------------------------------------------------------------------------------
               model           Beta    Std. Error    Std. Beta       t        Sig           lower          upper 
-----------------------------------------------------------------------------------------------------------------
         (Intercept)     543071.420    136210.918                   3.987    0.000     275874.535     810268.305 
            AREA_SQM      12688.669       370.119        0.579     34.283    0.000      11962.627      13414.710 
                 AGE     -24566.001      2766.041       -0.166     -8.881    0.000     -29991.980     -19140.022 
            PROX_CBD     -78121.985      6791.377       -0.267    -11.503    0.000     -91444.227     -64799.744 
      PROX_CHILDCARE    -333219.036    111020.303       -0.087     -3.001    0.003    -551000.984    -115437.089 
    PROX_ELDERLYCARE     170949.961     42110.748        0.083      4.060    0.000      88343.803     253556.120 
PROX_URA_GROWTH_AREA      38507.622     12523.661        0.059      3.075    0.002      13940.700      63074.545 
  PROX_HAWKER_MARKET      23801.197     29299.923        0.019      0.812    0.417     -33674.725      81277.120 
   PROX_KINDERGARTEN     144097.972     82738.669        0.030      1.742    0.082     -18205.570     306401.514 
            PROX_MRT    -322775.874     58528.079       -0.123     -5.515    0.000    -437586.937    -207964.811 
           PROX_PARK     564487.876     66563.011        0.148      8.481    0.000     433915.162     695060.590 
    PROX_PRIMARY_SCH     186170.524     65515.193        0.072      2.842    0.005      57653.253     314687.795 
PROX_TOP_PRIMARY_SCH       -477.073     20597.972       -0.001     -0.023    0.982     -40882.894      39928.747 
  PROX_SHOPPING_MALL    -207721.520     42855.500       -0.109     -4.847    0.000    -291788.613    -123654.427 
    PROX_SUPERMARKET     -48074.679     77145.257       -0.012     -0.623    0.533    -199405.956     103256.599 
       PROX_BUS_STOP     675755.044    138551.991        0.133      4.877    0.000     403965.817     947544.272 
         NO_Of_UNITS       -216.180        90.302       -0.046     -2.394    0.017       -393.320        -39.040 
     FAMILY_FRIENDLY     142128.272     47055.082        0.056      3.020    0.003      49823.107     234433.438 
            FREEHOLD     300646.543     77296.529        0.117      3.890    0.000     149018.525     452274.561 
      LEASEHOLD_99YR     -77137.375     77570.869       -0.030     -0.994    0.320    -229303.551      75028.801 
-----------------------------------------------------------------------------------------------------------------

3.4 Checking Multicollinearity

We use the following code to compute the Variance Inflation Factor (VIF) to help us identify multicollinearity. A VIF between 5 and 10 indicates moderate multicollinearity and requires monitoring, while a VIF above 10 suggests severe multicollinearity, warranting variable elimination.

As all VIF values are below 10, no variables need to be removed.

ols_vif_tol(condo_mlr)
              Variables Tolerance      VIF
1              AREA_SQM 0.8601326 1.162611
2                   AGE 0.7011585 1.426211
3              PROX_CBD 0.4575471 2.185567
4        PROX_CHILDCARE 0.2898233 3.450378
5      PROX_ELDERLYCARE 0.5922238 1.688551
6  PROX_URA_GROWTH_AREA 0.6614081 1.511926
7    PROX_HAWKER_MARKET 0.4373874 2.286303
8     PROX_KINDERGARTEN 0.8356793 1.196631
9              PROX_MRT 0.4949877 2.020252
10            PROX_PARK 0.8015728 1.247547
11     PROX_PRIMARY_SCH 0.3823248 2.615577
12 PROX_TOP_PRIMARY_SCH 0.4878620 2.049760
13   PROX_SHOPPING_MALL 0.4903052 2.039546
14     PROX_SUPERMARKET 0.6142127 1.628100
15        PROX_BUS_STOP 0.3311024 3.020213
16          NO_Of_UNITS 0.6543336 1.528272
17      FAMILY_FRIENDLY 0.7191719 1.390488
18             FREEHOLD 0.2728521 3.664990
19       LEASEHOLD_99YR 0.2645988 3.779307

3.5 Variable Selection

We use the ols_step_forward_p() function to perform stepwise forward selection. Although there are other criteria that could guide the selection process, we prioritise the p-value to ensure that all variables included in the final model are statistically significant.

We can visualise the stepwise forward selection process using the plot() function, which displays the incremental changes in Adjusted R², AIC, and RMSE throughout the selection process.

condo_fw_mlr <- ols_step_forward_p(condo_mlr,
                                   p_val = 0.05,
                                   details = FALSE)
plot(condo_fw_mlr)

3.6 Visualising Model Parameters

The following method allows us to visualise all the model parameters.

ggcoefstats(condo_mlr, sort = "ascending")

3.7 Testing for Non-linearity

It is important to test the assumption of linearity and additivity in the relationship between the dependent and independent variables. The figure shows that most data points are scattered around the zero line, indicating that the relationships between the dependent and independent variables are linear.

ols_plot_resid_fit(condo_fw_mlr$model)

3.8 Testing Normality of Residuals

We use ols_plot_resid_hist() and ols_test_normality() to check the normality of the residuals.

ols_plot_resid_hist(condo_fw_mlr$model)

ols_test_normality(condo_fw_mlr$model)
-----------------------------------------------
       Test             Statistic       pvalue  
-----------------------------------------------
Shapiro-Wilk              0.6856         0.0000 
Kolmogorov-Smirnov        0.1366         0.0000 
Cramer-von Mises         121.0768        0.0000 
Anderson-Darling         67.9551         0.0000 
-----------------------------------------------

3.9 Testing for Spatial Autocorrelation

The hedonic model incorporates geographically referenced attributes, making it essential to visualize the residuals. To test for spatial autocorrelation, we must convert the condo_resale.sf dataset from an sf object to a SpatialPointsDataFrame.

First, we export the residuals from the hedonic pricing model and save them as a data frame.

mlr_output <- as.data.frame(condo_fw_mlr$model$residuals) %>%
  rename(`FW_MLR_RES` = `condo_fw_mlr$model$residuals`)

Next, we will join the newly created dataframe with condo_resale_sf object.

condo_resale_sf <- cbind(condo_resale_sf,
                         mlr_output$FW_MLR_RES) %>%
  rename(`MLR_RES` = `mlr_output.FW_MLR_RES`)

3.10 Map Plot of Residuals

We plot the residuals on a map to identify areas of overestimation and underestimation. Visible clusters of over- or under-estimated prices may indicate the presence of spatial autocorrelation.

tmap_mode("view")
#tmap_options(check.and.fix = TRUE) -- add this code here to fix any layers with problematic lines/polygons.

tm_shape(mpsz) +
  tmap_options(check.and.fix = TRUE) + # add this line here to explicitly fix problematic polygons in this specific layer.
  tm_polygons(alpha = 0.4) +
  tm_shape(condo_resale_sf) +  
  tm_dots(col = "MLR_RES",
          alpha = 0.6,
          style="quantile") +
  tm_view(set.zoom.limits = c(11,14))
tmap_mode("plot")

3.11 Spatial Stationarity Test

We compute the distance-based weight matrix using sfdep package.

condo_resale_sf <- condo_resale_sf %>%
  mutate(nb = st_knn(geometry, k = 6, longlat = FALSE),
         wt = st_weights(nb, style = "W"),
         .before = 1)

To confirm the presence of spatial autocorrelation, we perform the Moran’s I test.

  • H₀: The residuals are randomly distributed (spatially stationary).
  • H₁: The residuals exhibit spatial dependence (spatially non-stationary).

We conduct a Global Moran’s I permutation test to determine whether spatial autocorrelation exists in the residuals.

The Global Moran’s I test for residual spatial autocorrelation shows that it’s p-value is less than the alpha value of 0.05. Hence, we will reject the null hypothesis that the residuals are randomly distributed. Since the Observed Global Moran I = 0.25586 which is greater than 0, we can infer than the residuals resemble cluster distribution.

global_moran_perm(condo_resale_sf$MLR_RES,
                  condo_resale_sf$nb,
                  condo_resale_sf$wt,
                  alternative = "two.sided",
                  nsim = 99)

    Monte-Carlo simulation of Moran I

data:  x 
weights: listw  
number of simulations + 1: 100 

statistic = 0.32254, observed rank = 100, p-value < 2.2e-16
alternative hypothesis: two.sided

4. Building GWR Model

In this section, we will build the hedonic pricing models using GWR.

4.1 Fixed Bandwidth GWR Model

In the code below, the bw.gwr() function from the GWmodel package is used to determine the optimal fixed bandwidth for the model. Setting the adaptive argument to FALSE specifies that a fixed bandwidth will be used.

There are two methods to define the stopping rule: the cross-validation (CV) approach and the AIC corrected (AICc) approach. We define the stopping rule using approach agreement.

bw_fixed <- bw.gwr(formula = SELLING_PRICE ~ AREA_SQM + AGE +
                     PROX_CBD + PROX_CHILDCARE + 
                     PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + 
                     PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + 
                     PROX_SHOPPING_MALL + PROX_BUS_STOP + 
                     NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                   data=condo_resale_sf,
                   approach="CV",
                   kernel = "gaussian",
                   adaptive = FALSE,
                   longlat = FALSE)
Fixed bandwidth: 17660.96 CV score: 8.259118e+14 
Fixed bandwidth: 10917.26 CV score: 7.970454e+14 
Fixed bandwidth: 6749.419 CV score: 7.273273e+14 
Fixed bandwidth: 4173.553 CV score: 6.300006e+14 
Fixed bandwidth: 2581.58 CV score: 5.404958e+14 
Fixed bandwidth: 1597.687 CV score: 4.857515e+14 
Fixed bandwidth: 989.6077 CV score: 4.722431e+14 
Fixed bandwidth: 613.7939 CV score: 1.379526e+16 
Fixed bandwidth: 1221.873 CV score: 4.778717e+14 
Fixed bandwidth: 846.0596 CV score: 4.791629e+14 
Fixed bandwidth: 1078.325 CV score: 4.751406e+14 
Fixed bandwidth: 934.7772 CV score: 4.72518e+14 
Fixed bandwidth: 1023.495 CV score: 4.730305e+14 
Fixed bandwidth: 968.6643 CV score: 4.721317e+14 
Fixed bandwidth: 955.7206 CV score: 4.722072e+14 
Fixed bandwidth: 976.6639 CV score: 4.721387e+14 
Fixed bandwidth: 963.7202 CV score: 4.721484e+14 
Fixed bandwidth: 971.7199 CV score: 4.721293e+14 
Fixed bandwidth: 973.6083 CV score: 4.721309e+14 
Fixed bandwidth: 970.5527 CV score: 4.721295e+14 
Fixed bandwidth: 972.4412 CV score: 4.721296e+14 
Fixed bandwidth: 971.2741 CV score: 4.721292e+14 
Fixed bandwidth: 970.9985 CV score: 4.721293e+14 
Fixed bandwidth: 971.4443 CV score: 4.721292e+14 
Fixed bandwidth: 971.5496 CV score: 4.721293e+14 
Fixed bandwidth: 971.3793 CV score: 4.721292e+14 
Fixed bandwidth: 971.3391 CV score: 4.721292e+14 
Fixed bandwidth: 971.3143 CV score: 4.721292e+14 
Fixed bandwidth: 971.3545 CV score: 4.721292e+14 
Fixed bandwidth: 971.3296 CV score: 4.721292e+14 
Fixed bandwidth: 971.345 CV score: 4.721292e+14 
Fixed bandwidth: 971.3355 CV score: 4.721292e+14 
Fixed bandwidth: 971.3413 CV score: 4.721292e+14 
Fixed bandwidth: 971.3377 CV score: 4.721292e+14 
Fixed bandwidth: 971.34 CV score: 4.721292e+14 
Fixed bandwidth: 971.3405 CV score: 4.721292e+14 
Fixed bandwidth: 971.3396 CV score: 4.721292e+14 
Fixed bandwidth: 971.3402 CV score: 4.721292e+14 
Fixed bandwidth: 971.3398 CV score: 4.721292e+14 
Fixed bandwidth: 971.34 CV score: 4.721292e+14 
Fixed bandwidth: 971.3399 CV score: 4.721292e+14 
Fixed bandwidth: 971.34 CV score: 4.721292e+14 

We use the code chunk below to build the fixed bandwidth GWR model. There are observable improvements in R2 and the AICc. (Note: AICc is robust for small dataset.)

gwr_fixed <- gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE +
                         PROX_CBD + PROX_CHILDCARE + 
                         PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + 
                         PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + 
                         PROX_SHOPPING_MALL + PROX_BUS_STOP + 
                         NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                       data=condo_resale_sf, 
                       bw=bw_fixed, 
                       kernel = 'gaussian', 
                       longlat = FALSE)
gwr_fixed
   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2024-10-16 23:56:50.289106 
   Call:
   gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
    PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + 
    PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + 
    PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
    data = condo_resale_sf, bw = bw_fixed, kernel = "gaussian", 
    longlat = FALSE)

   Dependent (y) variable:  SELLING_PRICE
   Independent variables:  AREA_SQM AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE PROX_URA_GROWTH_AREA PROX_MRT PROX_PARK PROX_PRIMARY_SCH PROX_SHOPPING_MALL PROX_BUS_STOP NO_Of_UNITS FAMILY_FRIENDLY FREEHOLD
   Number of data points: 1436
   ***********************************************************************
   *                    Results of Global Regression                     *
   ***********************************************************************

   Call:
    lm(formula = formula, data = data)

   Residuals:
     Min       1Q   Median       3Q      Max 
-3470778  -298119   -23481   248917 12234210 

   Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
   (Intercept)           527633.22  108183.22   4.877 1.20e-06 ***
   AREA_SQM               12777.52     367.48  34.771  < 2e-16 ***
   AGE                   -24687.74    2754.84  -8.962  < 2e-16 ***
   PROX_CBD              -77131.32    5763.12 -13.384  < 2e-16 ***
   PROX_CHILDCARE       -318472.75  107959.51  -2.950 0.003231 ** 
   PROX_ELDERLYCARE      185575.62   39901.86   4.651 3.61e-06 ***
   PROX_URA_GROWTH_AREA   39163.25   11754.83   3.332 0.000885 ***
   PROX_MRT             -294745.11   56916.37  -5.179 2.56e-07 ***
   PROX_PARK             570504.81   65507.03   8.709  < 2e-16 ***
   PROX_PRIMARY_SCH      159856.14   60234.60   2.654 0.008046 ** 
   PROX_SHOPPING_MALL   -220947.25   36561.83  -6.043 1.93e-09 ***
   PROX_BUS_STOP         682482.22  134513.24   5.074 4.42e-07 ***
   NO_Of_UNITS             -245.48      87.95  -2.791 0.005321 ** 
   FAMILY_FRIENDLY       146307.58   46893.02   3.120 0.001845 ** 
   FREEHOLD              350599.81   48506.48   7.228 7.98e-13 ***

   ---Significance stars
   Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
   Residual standard error: 756000 on 1421 degrees of freedom
   Multiple R-squared: 0.6507
   Adjusted R-squared: 0.6472 
   F-statistic: 189.1 on 14 and 1421 DF,  p-value: < 2.2e-16 
   ***Extra Diagnostic information
   Residual sum of squares: 8.120609e+14
   Sigma(hat): 752522.9
   AIC:  42966.76
   AICc:  42967.14
   BIC:  41731.39
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Fixed bandwidth: 971.34 
   Regression points: the same locations as observations are used.
   Distance metric: Euclidean distance metric is used.

   ****************Summary of GWR coefficient estimates:******************
                               Min.     1st Qu.      Median     3rd Qu.
   Intercept            -3.5988e+07 -5.1998e+05  7.6780e+05  1.7412e+06
   AREA_SQM              1.0003e+03  5.2758e+03  7.4740e+03  1.2301e+04
   AGE                  -1.3475e+05 -2.0813e+04 -8.6260e+03 -3.7784e+03
   PROX_CBD             -7.7047e+07 -2.3608e+05 -8.3599e+04  3.4646e+04
   PROX_CHILDCARE       -6.0097e+06 -3.3667e+05 -9.7426e+04  2.9007e+05
   PROX_ELDERLYCARE     -3.5001e+06 -1.5970e+05  3.1970e+04  1.9577e+05
   PROX_URA_GROWTH_AREA -3.0170e+06 -8.2013e+04  7.0749e+04  2.2612e+05
   PROX_MRT             -3.5282e+06 -6.5836e+05 -1.8833e+05  3.6922e+04
   PROX_PARK            -1.2062e+06 -2.1732e+05  3.5383e+04  4.1335e+05
   PROX_PRIMARY_SCH     -2.2695e+07 -1.7066e+05  4.8472e+04  5.1555e+05
   PROX_SHOPPING_MALL   -7.2585e+06 -1.6684e+05 -1.0517e+04  1.5923e+05
   PROX_BUS_STOP        -1.4676e+06 -4.5207e+04  3.7601e+05  1.1664e+06
   NO_Of_UNITS          -1.3170e+03 -2.4822e+02 -3.0846e+01  2.5496e+02
   FAMILY_FRIENDLY      -2.2749e+06 -1.1140e+05  7.6214e+03  1.6107e+05
   FREEHOLD             -9.2067e+06  3.8074e+04  1.5169e+05  3.7528e+05
                             Max.
   Intercept            112794435
   AREA_SQM                 21575
   AGE                     434203
   PROX_CBD               2704604
   PROX_CHILDCARE         1654086
   PROX_ELDERLYCARE      38867861
   PROX_URA_GROWTH_AREA  78515805
   PROX_MRT               3124325
   PROX_PARK             18122439
   PROX_PRIMARY_SCH       4637517
   PROX_SHOPPING_MALL     1529953
   PROX_BUS_STOP         11342209
   NO_Of_UNITS              12907
   FAMILY_FRIENDLY        1720745
   FREEHOLD               6073642
   ************************Diagnostic information*************************
   Number of data points: 1436 
   Effective number of parameters (2trace(S) - trace(S'S)): 438.3807 
   Effective degrees of freedom (n-2trace(S) + trace(S'S)): 997.6193 
   AICc (GWR book, Fotheringham, et al. 2002, p. 61, eq 2.33): 42263.61 
   AIC (GWR book, Fotheringham, et al. 2002,GWR p. 96, eq. 4.22): 41632.36 
   BIC (GWR book, Fotheringham, et al. 2002,GWR p. 61, eq. 2.34): 42515.71 
   Residual sum of squares: 2.534069e+14 
   R-square value:  0.8909912 
   Adjusted R-square value:  0.8430418 

   ***********************************************************************
   Program stops at: 2024-10-16 23:56:50.824826 

4.2 Adaptive Bandwidth GWR Model

The following code is similar to the one used for computing fixed bandwidth, except that the adaptive argument is set to TRUE. The recommended number of data points to use is 30.

bw_adaptive <- bw.gwr(formula = SELLING_PRICE ~ AREA_SQM + AGE +
                     PROX_CBD + PROX_CHILDCARE + 
                     PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + 
                     PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + 
                     PROX_SHOPPING_MALL + PROX_BUS_STOP + 
                     NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                   data=condo_resale_sf,
                   approach="CV",
                   kernel = "gaussian",
                   adaptive = TRUE,
                   longlat = FALSE)
Adaptive bandwidth: 895 CV score: 7.952401e+14 
Adaptive bandwidth: 561 CV score: 7.667364e+14 
Adaptive bandwidth: 354 CV score: 6.953454e+14 
Adaptive bandwidth: 226 CV score: 6.15223e+14 
Adaptive bandwidth: 147 CV score: 5.674373e+14 
Adaptive bandwidth: 98 CV score: 5.426745e+14 
Adaptive bandwidth: 68 CV score: 5.168117e+14 
Adaptive bandwidth: 49 CV score: 4.859631e+14 
Adaptive bandwidth: 37 CV score: 4.646518e+14 
Adaptive bandwidth: 30 CV score: 4.422088e+14 
Adaptive bandwidth: 25 CV score: 4.430816e+14 
Adaptive bandwidth: 32 CV score: 4.505602e+14 
Adaptive bandwidth: 27 CV score: 4.462172e+14 
Adaptive bandwidth: 30 CV score: 4.422088e+14 

We use the code chunk below to build the adaptive bandwidth GWR model. The report shows that the AICc the adaptive distance gwr is 41982.22 which is even smaller than the AICc of the fixed distance gwr of 42263.61.

gwr_adaptive <- gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE +
                            PROX_CBD + PROX_CHILDCARE + 
                            PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + 
                            PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + 
                            PROX_SHOPPING_MALL + PROX_BUS_STOP + 
                            NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
                         data=condo_resale_sf, 
                         bw=bw_adaptive,
                         kernel = 'gaussian', 
                         adaptive=TRUE,
                         longlat = FALSE)
gwr_adaptive
   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2024-10-16 23:56:55.416896 
   Call:
   gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + 
    PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + 
    PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + 
    PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, 
    data = condo_resale_sf, bw = bw_adaptive, kernel = "gaussian", 
    adaptive = TRUE, longlat = FALSE)

   Dependent (y) variable:  SELLING_PRICE
   Independent variables:  AREA_SQM AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE PROX_URA_GROWTH_AREA PROX_MRT PROX_PARK PROX_PRIMARY_SCH PROX_SHOPPING_MALL PROX_BUS_STOP NO_Of_UNITS FAMILY_FRIENDLY FREEHOLD
   Number of data points: 1436
   ***********************************************************************
   *                    Results of Global Regression                     *
   ***********************************************************************

   Call:
    lm(formula = formula, data = data)

   Residuals:
     Min       1Q   Median       3Q      Max 
-3470778  -298119   -23481   248917 12234210 

   Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
   (Intercept)           527633.22  108183.22   4.877 1.20e-06 ***
   AREA_SQM               12777.52     367.48  34.771  < 2e-16 ***
   AGE                   -24687.74    2754.84  -8.962  < 2e-16 ***
   PROX_CBD              -77131.32    5763.12 -13.384  < 2e-16 ***
   PROX_CHILDCARE       -318472.75  107959.51  -2.950 0.003231 ** 
   PROX_ELDERLYCARE      185575.62   39901.86   4.651 3.61e-06 ***
   PROX_URA_GROWTH_AREA   39163.25   11754.83   3.332 0.000885 ***
   PROX_MRT             -294745.11   56916.37  -5.179 2.56e-07 ***
   PROX_PARK             570504.81   65507.03   8.709  < 2e-16 ***
   PROX_PRIMARY_SCH      159856.14   60234.60   2.654 0.008046 ** 
   PROX_SHOPPING_MALL   -220947.25   36561.83  -6.043 1.93e-09 ***
   PROX_BUS_STOP         682482.22  134513.24   5.074 4.42e-07 ***
   NO_Of_UNITS             -245.48      87.95  -2.791 0.005321 ** 
   FAMILY_FRIENDLY       146307.58   46893.02   3.120 0.001845 ** 
   FREEHOLD              350599.81   48506.48   7.228 7.98e-13 ***

   ---Significance stars
   Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
   Residual standard error: 756000 on 1421 degrees of freedom
   Multiple R-squared: 0.6507
   Adjusted R-squared: 0.6472 
   F-statistic: 189.1 on 14 and 1421 DF,  p-value: < 2.2e-16 
   ***Extra Diagnostic information
   Residual sum of squares: 8.120609e+14
   Sigma(hat): 752522.9
   AIC:  42966.76
   AICc:  42967.14
   BIC:  41731.39
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Adaptive bandwidth: 30 (number of nearest neighbours)
   Regression points: the same locations as observations are used.
   Distance metric: Euclidean distance metric is used.

   ****************Summary of GWR coefficient estimates:******************
                               Min.     1st Qu.      Median     3rd Qu.
   Intercept            -1.3487e+08 -2.4669e+05  7.7928e+05  1.6194e+06
   AREA_SQM              3.3188e+03  5.6285e+03  7.7825e+03  1.2738e+04
   AGE                  -9.6746e+04 -2.9288e+04 -1.4043e+04 -5.6119e+03
   PROX_CBD             -2.5330e+06 -1.6256e+05 -7.7242e+04  2.6624e+03
   PROX_CHILDCARE       -1.2790e+06 -2.0175e+05  8.7158e+03  3.7778e+05
   PROX_ELDERLYCARE     -1.6212e+06 -9.2050e+04  6.1029e+04  2.8184e+05
   PROX_URA_GROWTH_AREA -7.2686e+06 -3.0350e+04  4.5869e+04  2.4613e+05
   PROX_MRT             -4.3781e+07 -6.7282e+05 -2.2115e+05 -7.4593e+04
   PROX_PARK            -2.9020e+06 -1.6782e+05  1.1601e+05  4.6572e+05
   PROX_PRIMARY_SCH     -8.6418e+05 -1.6627e+05 -7.7853e+03  4.3222e+05
   PROX_SHOPPING_MALL   -1.8272e+06 -1.3175e+05 -1.4049e+04  1.3799e+05
   PROX_BUS_STOP        -2.0579e+06 -7.1461e+04  4.1104e+05  1.2071e+06
   NO_Of_UNITS          -2.1993e+03 -2.3685e+02 -3.4699e+01  1.1657e+02
   FAMILY_FRIENDLY      -5.9879e+05 -5.0927e+04  2.6173e+04  2.2481e+05
   FREEHOLD             -1.6340e+05  4.0765e+04  1.9023e+05  3.7960e+05
                            Max.
   Intercept            18758355
   AREA_SQM                23064
   AGE                     13303
   PROX_CBD             11346650
   PROX_CHILDCARE        2892127
   PROX_ELDERLYCARE      2465671
   PROX_URA_GROWTH_AREA  7384059
   PROX_MRT              1186242
   PROX_PARK             2588497
   PROX_PRIMARY_SCH      3381462
   PROX_SHOPPING_MALL   38038564
   PROX_BUS_STOP        12081592
   NO_Of_UNITS              1010
   FAMILY_FRIENDLY       2072414
   FREEHOLD              1813995
   ************************Diagnostic information*************************
   Number of data points: 1436 
   Effective number of parameters (2trace(S) - trace(S'S)): 350.3088 
   Effective degrees of freedom (n-2trace(S) + trace(S'S)): 1085.691 
   AICc (GWR book, Fotheringham, et al. 2002, p. 61, eq 2.33): 41982.22 
   AIC (GWR book, Fotheringham, et al. 2002,GWR p. 96, eq. 4.22): 41546.74 
   BIC (GWR book, Fotheringham, et al. 2002,GWR p. 61, eq. 2.34): 41914.08 
   Residual sum of squares: 2.528227e+14 
   R-square value:  0.8912425 
   Adjusted R-square value:  0.8561185 

   ***********************************************************************
   Program stops at: 2024-10-16 23:56:56.247667 

4.3 Visualising SDF Fields

To visualise the fields in SDF, we need to first covert it into sf data.frame by using the code chunk below.

gwr_adaptive_output <- as.data.frame(gwr_adaptive$SDF) %>%
  select(-c(2:15))

gwr_sf_adaptive <- cbind(condo_resale_sf,
                         gwr_adaptive_output)
glimpse(gwr_sf_adaptive)
Rows: 1,436
Columns: 63
$ nb                      <nb> <66, 77, 123, 238, 239, 343>, <21, 162, 163, 19…
$ wt                      <list> <0.1666667, 0.1666667, 0.1666667, 0.1666667, …
$ POSTCODE                <dbl> 118635, 288420, 267833, 258380, 467169, 466472…
$ SELLING_PRICE           <dbl> 3000000, 3880000, 3325000, 4250000, 1400000, 1…
$ AREA_SQM                <dbl> 309, 290, 248, 127, 145, 139, 218, 141, 165, 1…
$ AGE                     <dbl> 30, 32, 33, 7, 28, 22, 24, 24, 27, 31, 17, 22,…
$ PROX_CBD                <dbl> 7.941259, 6.609797, 6.898000, 4.038861, 11.783…
$ PROX_CHILDCARE          <dbl> 0.16597932, 0.28027246, 0.42922669, 0.39473543…
$ PROX_ELDERLYCARE        <dbl> 2.5198118, 1.9333338, 0.5021395, 1.9910316, 1.…
$ PROX_URA_GROWTH_AREA    <dbl> 6.618741, 7.505109, 6.463887, 4.906512, 6.4106…
$ PROX_HAWKER_MARKET      <dbl> 1.76542207, 0.54507614, 0.37789301, 1.68259969…
$ PROX_KINDERGARTEN       <dbl> 0.05835552, 0.61592412, 0.14120309, 0.38200076…
$ PROX_MRT                <dbl> 0.5607188, 0.6584461, 0.3053433, 0.6910183, 0.…
$ PROX_PARK               <dbl> 1.1710446, 0.1992269, 0.2779886, 0.9832843, 0.…
$ PROX_PRIMARY_SCH        <dbl> 1.6340256, 0.9747834, 1.4715016, 1.4546324, 0.…
$ PROX_TOP_PRIMARY_SCH    <dbl> 3.3273195, 0.9747834, 1.4715016, 2.3006394, 0.…
$ PROX_SHOPPING_MALL      <dbl> 2.2102717, 2.9374279, 1.2256850, 0.3525671, 1.…
$ PROX_SUPERMARKET        <dbl> 0.9103958, 0.5900617, 0.4135583, 0.4162219, 0.…
$ PROX_BUS_STOP           <dbl> 0.10336166, 0.28673408, 0.28504777, 0.29872340…
$ NO_Of_UNITS             <dbl> 18, 20, 27, 30, 30, 31, 32, 32, 32, 32, 34, 34…
$ FAMILY_FRIENDLY         <dbl> 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0…
$ FREEHOLD                <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1…
$ LEASEHOLD_99YR          <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ MLR_RES                 <dbl> -1489099.55, 415494.57, 194129.69, 1088992.71,…
$ Intercept               <dbl> 2050011.67, 1633128.24, 3433608.17, 234358.91,…
$ y                       <dbl> 3000000, 3880000, 3325000, 4250000, 1400000, 1…
$ yhat                    <dbl> 2886531.8, 3466801.5, 3616527.2, 5435481.6, 13…
$ residual                <dbl> 113468.16, 413198.52, -291527.20, -1185481.63,…
$ CV_Score                <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ Stud_residual           <dbl> 0.38207013, 1.01433140, -0.83780678, -2.846146…
$ Intercept_SE            <dbl> 516105.5, 488083.5, 963711.4, 444185.5, 211962…
$ AREA_SQM_SE             <dbl> 823.2860, 825.2380, 988.2240, 617.4007, 1376.2…
$ AGE_SE                  <dbl> 5889.782, 6226.916, 6510.236, 6010.511, 8180.3…
$ PROX_CBD_SE             <dbl> 37411.22, 23615.06, 56103.77, 469337.41, 41064…
$ PROX_CHILDCARE_SE       <dbl> 319111.1, 299705.3, 349128.5, 304965.2, 698720…
$ PROX_ELDERLYCARE_SE     <dbl> 120633.34, 84546.69, 129687.07, 127150.69, 327…
$ PROX_URA_GROWTH_AREA_SE <dbl> 56207.39, 76956.50, 95774.60, 470762.12, 47433…
$ PROX_MRT_SE             <dbl> 185181.3, 281133.9, 275483.7, 279877.1, 363830…
$ PROX_PARK_SE            <dbl> 205499.6, 229358.7, 314124.3, 227249.4, 364580…
$ PROX_PRIMARY_SCH_SE     <dbl> 152400.7, 165150.7, 196662.6, 240878.9, 249087…
$ PROX_SHOPPING_MALL_SE   <dbl> 109268.8, 98906.8, 119913.3, 177104.1, 301032.…
$ PROX_BUS_STOP_SE        <dbl> 600668.6, 410222.1, 464156.7, 562810.8, 740922…
$ NO_Of_UNITS_SE          <dbl> 218.1258, 208.9410, 210.9828, 361.7767, 299.50…
$ FAMILY_FRIENDLY_SE      <dbl> 131474.73, 114989.07, 146607.22, 108726.62, 16…
$ FREEHOLD_SE             <dbl> 115954.0, 130110.0, 141031.5, 138239.1, 210641…
$ Intercept_TV            <dbl> 3.9720784, 3.3460017, 3.5629010, 0.5276150, 1.…
$ AREA_SQM_TV             <dbl> 11.614302, 20.087361, 13.247868, 33.577223, 4.…
$ AGE_TV                  <dbl> -1.6154474, -9.3441881, -4.1023685, -15.524301…
$ PROX_CBD_TV             <dbl> -3.22582173, -6.32792021, -4.62353528, 5.17080…
$ PROX_CHILDCARE_TV       <dbl> 1.000488185, 1.471786337, -0.344047555, 1.5766…
$ PROX_ELDERLYCARE_TV     <dbl> -3.26126929, 3.84626245, 4.13191383, 2.4756745…
$ PROX_URA_GROWTH_AREA_TV <dbl> -2.846248368, -1.848971738, -2.648105057, -5.6…
$ PROX_MRT_TV             <dbl> -1.61864578, -8.92998600, -3.40075727, -7.2870…
$ PROX_PARK_TV            <dbl> -0.83749312, 2.28192684, 0.66565951, -3.340617…
$ PROX_PRIMARY_SCH_TV     <dbl> 1.59230221, 6.70194543, 2.90580089, 12.9836104…
$ PROX_SHOPPING_MALL_TV   <dbl> 2.753588422, -0.886626400, -1.056869486, -0.16…
$ PROX_BUS_STOP_TV        <dbl> 2.0154464, 4.4941192, 3.0419145, 12.8383775, 0…
$ NO_Of_UNITS_TV          <dbl> 0.480589953, -1.380026395, -0.045279967, -0.44…
$ FAMILY_FRIENDLY_TV      <dbl> -0.06902748, 2.69655779, 0.04058290, 14.312764…
$ FREEHOLD_TV             <dbl> 2.6213469, 3.0452799, 1.1970499, 8.7711485, 1.…
$ Local_R2                <dbl> 0.8846744, 0.8899773, 0.8947007, 0.9073605, 0.…
$ geometry                <POINT [m]> POINT (22085.12 29951.54), POINT (25656.…
$ geometry.1              <POINT [m]> POINT (22085.12 29951.54), POINT (25656.…

The code chunks below is used to create an interactive point symbol map of local R2

tmap_mode("view")
tm_shape(mpsz)+
  tm_polygons(alpha = 0.1) +
tm_shape(gwr_sf_adaptive) +  
  tm_dots(col = "Local_R2",
          border.col = "gray60",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(11,14))
tmap_mode("plot")
tmap_options(check.and.fix = TRUE)
tmap_mode("view")
AREA_SQM_SE <- tm_shape(mpsz)+
  tm_polygons(alpha = 0.1) +
tm_shape(gwr_sf_adaptive) +  
  tm_dots(col = "AREA_SQM_SE",
          border.col = "gray60",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(11,14))

AREA_SQM_TV <- tm_shape(mpsz)+
  tm_polygons(alpha = 0.1) +
tm_shape(gwr_sf_adaptive) +  
  tm_dots(col = "AREA_SQM_TV",
          border.col = "gray60",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(11,14))

tmap_arrange(AREA_SQM_SE, AREA_SQM_TV, 
             asp=1, ncol=2,
             sync = TRUE)
tm_shape(mpsz[mpsz$REGION_N=="CENTRAL REGION", ])+
  tm_polygons()+
tm_shape(gwr_sf_adaptive) + 
  tm_bubbles(col = "Local_R2",
           size = 0.15,
           border.col = "gray60",
           border.lwd = 1)